# The Seifert-van Kampen Theorem

We are now ready to state a very important theorem for determining the fundamental groups of certain spaces. This theorem is called the Seifert-van Kampen theorem. We first state the simplest version of the theorem and then state a more general version.

Theorem 1 (The Seifert-van Kampen Theorem): Let $X$ be a topological space and let $\{ U_1, U_2 \}$ be an open cover of $X$. Let $\pi_1(U_1, x) = \langle A_1: \mathcal R_1 \rangle$ and $\pi_1(U_2, x) = \langle A_2 : \mathcal R_2 \rangle$. If:1) $U_1$, $U_2$, and $U_1 \cap U_2$ are path connected.2) $x \in U_1 \cap U_2$.Then $\pi_1(X, x)$ is the free product of $\pi_1(U_1, x)$ and $\pi_1(U_2, x)$ with amalgamation by $\pi_1(U_1 \cap U_2, x)$ by $i_{1*} : \pi_1(U_1 \cap U_2, x) \to \pi_1(U_1, x)$ and $i_{2*} : \pi_1(U_1 \cap U_2, x) \to \pi_1(U_2, x)$, that is, $\pi_1(X, x) = \langle A_1 \cup A_2 : \mathcal R_1 \cup \mathcal R_2 \cup \{ i_{1*}(h) = i_{2^*}(h) : h \in \pi_1(U_1 \cap U_2, x) \} \rangle$. |

*We denote the inclusion map of $U_1 \cap U_2$ in $U_1$ by $i_1 : U_1 \cap U_2 \to U_1$, and we denote the inclusion map of $U_1 \cap U_2$ in $U_2$ by $i_2 : U_1 \cap U_2 \to U_2$. Hence, $i_{1*} : \pi_1(U_1 \cap U_2, x) \to \pi_1(U_1, x)$ and $i_{2*} : \pi_1(U_1 \cap U_2, x) \to \pi_1(U_2, x)$ are the induced mapping as seen on The Induced Mapping from the Fundamental Groups of Two Topological Spaces page, and they are also group homomorphisms.*

Theorem 2 (The General Seifert-van Kampen Theorem): Let $X$ be a topological space and let $\{ U_j : j \in J \}$ be an open cover of $X$. Let $\pi_1(U_j, x) = \langle A_j, \mathcal R_j \rangle$ for each $j \in J$. If:1) Each intersection of four or less $U_j$s is path connected.2) $\displaystyle{x \in \bigcap_{j \in J} U_j}$.Then $\pi_1(X, x)$ is the free product of the $\pi_1(U_j, x)$s with amalgamation by the maps $\phi_{ij*}^{i} : \pi_1(U_i \cap U_j, x) \to \pi_1(U_i, x)$ and $\phi_{ij*}^{j} : \pi_1(U_i \cap U_j, x) \to \pi_1(U_j, x)$, that is, $\pi_1(X, x) = \biggr \langle \bigcup_{j \in J} A_j : \bigcup_{j \in J} \mathcal R_j \cup \bigcup_{i, j \in J} \{ \phi_{ij*}^{i}(h) = \phi_{ij*}^{j}(h) : h \in \pi_1(U_i \cap U_j, x) \} \biggr \rangle$. |

*Here, $\phi_{ij}^{i} : U_i \cap U_j \to U_i$ and $\phi_{ij}^{j} : U_i \cap U_j \to U_j$ are the usual inclusion maps.*

We look at some examples of applying the Seifert-van Kampen theorem on the following pages: